An isothermal expansion of a gas is such that the temperature of the gas is kept constant. The diagram below illustrates the isothermal expansion of a gas from
to
during which the pressure falls from
to
The gas expands during this expansion, so work is done by the gas. The amount of work done is equal to the area under the graph. We can write it as an integral:
If the gas is ideal then we can sue the ideal gas equation
rearranged as
to obtain the integral
We can evaluate this integral quite easily, obtaining
(1)
Because
the gas does work so W is positive.
Notice also that
for an isothermal process so that
This means we can write equation (1) as
